A Brief Summary of Kaluza-Klein Theory

Jonah Millerjonah.maxwell.miller@gmail.com

Friday May 3rd, 2013

Abstract

We briefly discuss the origins of Kaluza-Klein theory. First, we describe Kaluzas original cylindercondition and how this allows for the unification of Einstein gravity and electromagnetism in a five-dimensional framework. We then discuss possible justifications of the cylinder condition, with a specialemphasis on Kleins compactification approach.

1 Introduction and Some History

As early as 1914, physicists dreamed of a geometric unification the fundamental forces of nature. Inspiredby Minkowskis treatment of Maxwells equations in special relativity [1], Gunnar Nordstrom unified electro-magnetism and Newtonian gravity by embedding a symmetric energy-momentum tensor in five-dimensionalMinkowski space [2, 3]. Since Nordstroms idea relies on Newtonian gravity, its not particularly relevanttoday. But given that Nordstom published his idea before the discovery of general relativity, it does representa remarkably prescient geometric approach to fundamental physics.

Only seven years later, now after the discovery of general relativity, Theodor Kaluza discovered that themetric tensor of five-dimensional curved spacetime can be viewed as the metric of four-dimensional spacetimecoupled to the electromagnetic vector potential and a scalar field [2, 4]. He sent his idea to Einstein, whoencouraged him to publish [2] and who later expanded on Kaluzas idea [2, 5].

Kaluza-Klein theory gets the other half of its name from Oscar Klein. As we will see in section 2, Kaluzastheory imposes a rather arbitrary conditionwhich he called the cylinder conditionon the geometry.In an innovative attempt to quantize electric charge, Klein suggested one explanation for this condition,where the extra fifth dimension is periodic and very small. See section 3 for details. The insights of Kaluzaand Klein have inspired a plethora of higher-dimensional unification schemes, from supergravity and stringtheory [2, 6, 7] to Brane theory and higher-dimensional cosmologies [8, 9].

In this work, we discuss Kaluzas initial insight in section 2 and Kleins extension of the theory in 3.Kleins method is not the only physically compelling way to study the cylinder condition and we will brieflydiscuss the other approaches in that section as well. Finally, in section 4 we offer some concluding remarks.

2 Kaluzas Five-Dimensional Spacetime

We now present Kaluzas original idea with the arbitrary cylinder condition. Although Kaluza originallyused Einsteins tensor notation [4], Thiry demonstrated that the langauge of differential forms is better suitedto the task at hand [10], so we will use that notation. We borrow the setup from Einstein and Bergmann [5]and then loosely follow Pope, but with more detail [6].

2.1 General Construction

Let M(5) be a (4+1)-dimensional Lorentzian manifold and letx = xAA (1)

1

be a set of general coordinates onM(5).1 Without loss of generality, we choose x0 to be the basis element ina timelike direction, i to be three spatial coordinates, and 4 to the be the fourth spatial coordinate, whichwill describe the extra fifth dimension. By convention we will choose timelike directions to have negativesquare norm and spacelike directions to have positive square norm. We will call the five-dimensional metricgAB .

2

2.1.1 The Cylinder Condition

At this point, there is no geometric difference between any of the spacelike coordinates; i i {1, 2, 3}, and4 are indistinguishable. Kaluzas cylinder condition, however, will make them distinct, and we are now in aposition to state it. Kaluza (rather apologetically) enforces that the five-dimensional metric be independentof x4 [4, 2, 7]. In other words, the derivatives of the metric with respect to the fifth coordinate must vanish:

4gAB = 0. (2)

This is a statement about symmetry. The cylinder condition demands that the spacetime have a specificisometry,

x4 x4 + , R. (3)Furthermore, it enforces a specific parametrization where 4 is the Killing vector associated with our isometry.Mathematicians call parametrizations of this type Clairaut [11], and the existence a Clairaut parametriza-tion is not at all guaranteed. We are requiring that at least one spacelike Killing vector field exists in thisspacetime and, since most spacetimes lack any isometries, this condition is quite restrictive.

This Clairaut parametrization does allow us to define tensors on M(4), however. We can foliate ourspacetime by spacelike hypersurfaces, each orthogonal to the Killing vector field 5, such that these hyper-surfaces are related by a global isometry. We call any such hypersurface M(4). This foliation allows us towrite down the topology of M(5):

M(5) =M(4) M(1), (4)where M(1) is some one-dimensional Euclidean manifold [5]. The choice of hypersurface in a given foliationis arbitrary, since the five-metric will be identical on each one.

2.1.2 The Metric

We want to choose a geometrically natural parametrization and write down a metric. Before we do that,however, lets define a few quantities which we will use to build our line element. First, let the four-dimensional metric on a given hypersurface be g . Let A be a four-covector field and let be a scalarfield on M(4).

Although they wont have tensorial properties on the general five-dimensional spacetime, we can certainlyextend these quantities and define them on M(5). However, we must impose that

4A = 0, 4g = 0, and 4 = 0. (5)

Although these names are suggestiveA suggests a vector potential and suggests a scalar fieldtheycurrently contain no physical information. They will only take on physical meaning once we write down ametric and study the geometry of the space.

And now well do just that. We can choose a parametrization of the space so that the five-dimensionalline element takes the form

ds2 = e2gdxdx + e2(dx4 +Adx)2, (6)

1In this work, Greek indices will sum from zero to three and upper-case Latin indices from zero to four.2Later, we will define tensors on four-dimensional submanifolds ofM(5). We will denote five-dimensional tensors with hats

on them and four-dimensional tensors without the hats.

2

where and are currently arbitrary constants [6]. This means that the components of the five-dimensionalmetric tensor gAB take the form

g = e2g + e

2AA , g4 = e2A, and g44 = e2. (7)So long as 6= 0, this metric corresponds to a valid Clairaut parametrization of the space. Although thismetric looks a little funny, the reason for our choice will be clear soon.

To make the remaining calculations easier, lets transition to a coordinate-free description. Let be ea bea tetrad basis for one of the M(4) hypersurfaces such that

ab = eaeb g a, b {0, 1, 2, 3}.

We now want to write down a funfbein basis for the five dimensional spacetime, satisfying

ab = eAa e

Bb gAB a, b {0, 1, 2, 3, 4}.

Thanks to our nice choice of coordinates we can see by inspection3 that

ea = eea and ez = e(dx4 +Adx) (8)will work. Beware the abuse of notation! ea is a tetrad basis element, but e and e are Eulers constantraised to powers. In keeping with Pope, we will denote the coordinate-free basis elements by lower-case Latinindices. We set aside lower-case z for the basis element pointing in direction associated with the cylindercondition [6]. We will assume that that lower-case Latin indices range from zero to three. If they do not,the range will be indicated.

To calculate the curvature of the space, we turn to the Cartan structure equations [12],

T a = dea + ab eb, a, b {0, 1, 2, 3, 4}Rab = d

ab +

ac cb + az ez, a, b, c {0, 1, 2, 3, 4}

where T a, Rab, and ab are the five-dimensional torsion two-form, curvature two-form and spin connections

respectively. Since our spacetime is Torsion-free, the structure equations reduce to

0 = dea + ab eb + az ez (9)0 = dez + zb eb + zz ez (10)

Rab = dab +

ac cb + az ez (11)

Raz = daz +

ac cz + az ez (12)

Rzz = dzz +

zc cz + zz ez (13)

with our usual index conventions. Well need the following one-form,

d = AdxA = dx

. (14)

Lets use equations 9 and 10 to calculate the spin connection.

dea = d(eea

)= de ea + edeb= ed ea eab eb= ea(dx

ea) eaab eb,but, from equation 9,

dea = ab eb + az ez= eab eb eaz (dx4 +A)

3The power of working in a coordinate-free basis is that many quanities can be calculated by inspection.

3

too. So,ea(dx

ea) eaab eb = eab eb eaz (dx4 +A). (15)If we identify like terms,

ab eb = ab eb, (16)and e(e

a dx) = eaz (dx4 +A). (17)

Similarly,

dez = d[e(dx4 +A)]

= ed (dx4 +A) + ed2x4 + edA= edx

(dx4 +A) + eF , (18)

where F = dA. The d2x4 term disappears because d2 = 0 for any differential form (x4 is a zero-form).From equation 10,

dez = zb eb + zz ez= ezb eb + zz ez.

So, ezb eb = edx (dx4 +A) + eF . (19)

If we inspect equation 19 and re-absorb some of the scale factors into ez, we can infer that

az = za = eaez 12Fabe(2)eb, (20)

and zz = 0. (21)

where a = ea is the partial derivative translated into a coordinate-free basis with the help of the inverse

funfbeins on M(4), ea. Similarly, Fab denotes the components of F in the coordinate-free basis [6]. If weplug this result into equation 16, we find that

ab = ab + e(bea aeb) 12Fabe(2)ez. (22)

Its now relatively straightforward to compute the curvature two-form. For example, from equation 13,

Rzz = zc cz

= [eaez +

1

2Fabe(2)eb

][eaez +

1

2Face(2)ec

]= 2e2 (aez aez) 1

2Fab e(3)eb aez

1

2Fabe(3) (aez) eb

14e2(2)FabFaceb ec

= 2e2 (aez aez) + 12Fab e(3)eb aez

1

2Fab e(3)eb aez

14e2(2)FabFaceb ec

Rzz = 2e22+ 14e2(2)F2, (23)

whereF2 = FabFab,

4

and2 = a

a = (24)

is the Laplacian operator in four dimensions. The other calculations are similar. Pope gives the remainingterms:

Rab = e2 (Rab + ( 22)ab ab2) 1

2e2(2)F ca Fbc (25)

and Raz = Rza =1

2e()b

(e2(+)Fab

), (26)

where Rab is the curvature 2-form on M(4), ab is the four-dimensional Minkowski metric, and where b isthe covariant derivative on the coordinate-free basis [6].

Then the five-dimensional Ricci scalar comes from the trace of the curvature 2-form [6]:

R = abRab + Rzz

= e2(abRab + ( 22) ()2 aa2

) 1

2e2(2)F2 + 2e22+ 1

4e2(2)F2

= e2(R+ ( 22) ()2 + (2 2)2

) 1

4e2(2)F2. (27)

2.1.3 Lagrangian Density

We would like to calculate the Lagrangian density for the theory. First,g = e(+4)g. (28)

So,4

L =gR

=ge(+2) [R+ ( 22)()2 + (2 2)2] 1

4

ge3F2. (29)

We are now in a position to set and . We want our theory to include four-dimensional Einstein gravity,so the Lagrangian density better include a term that looks like

gR. This tells us that we want [6]:

= 2. (30)

If we do this, our Lagrangian density becomes [6]:

L = g[R 62()2 + (42 2)2 1

4e6F2

].

The term 62 ()2 is very reminiscent of the Klein-Gorden Lagrange density for a massless scalar field[6]. In the Klein-Gordon action, the term is 12 ()2 [6]. This informs our choice of [6]:

=112. (31)

With and set, we can write down our final Lagrangian density. Since the Laplacian term 2, whichgives a total derivative in L will have no effect on variational calculations, we drop it [6]:

L = g[R 1

2()

2 14e3F2

]. (32)

4For simplicity, we will work in units were 116piG

= 1.

5

From a five-dimensional theory that contains only gravity, we have constructed a four-dimensional La-grangian density that includes Einstein gravity and a scalar field coupled to an object that looks an awful lotlike the Maxwell tensor. Indeed, if we set = 0, we would recover the Einstein-Maxwell Lagrangian density,

L = g(R 1

4F2).

It would certainly be tempting to do sowe would get Einstein-Maxwell gravity and electromagnetism withno extra complications.5 Unfortunately, this is forbidden [6]. To see why, lets work out the field equations.

2.1.4 Field Equations

For simplicity, we will assume that the five-dimensional spacetime has no boundary and thus all boundaryterms will drop out. It is more convenient to use the variations g instead of g [14]. So,

g = ggg . (33)

We will also use the well-known variation of the determinant of the metric [14],

g = 1

2

ggg . (34)

Now, the variation of the action is

Skaluza =

d5xL

=

d5x

[g(R 12

()2 14e3F2

)] Skaluza =

g(R 12

()2 14e3F2

)d5x

+

(R 1

2()2 1

4e3F2

)gd5x. (35)

To make things simpler, well treat each integral separately. Let

S1 =

g(R 12

()2 14e3F2

)d5x (36)

and S2 =

(R 1

2()2 1

4e3F2

)gd5x. (37)

The first integral becomes

S1 =

g(R 12

()2 14e3F2

)d5x

=

g(gR 12g()() 1

4e3ggFF

)dx5

=

g [gR +Rg 12

()()g 1

2g()

]d5x

14

g [3F2+ 2se3FF g + 2e3FF] d5x. (38)5Historically, many people did set = 0likely because they distrusted scalar fields [2, 7]. However, neither Kaluza nor

Klein made this simplification [2, 4, 13, 6].

6

Similarly, if we use equation 34, the second integral becomes

S2 =

(R 1

2()2 1

4e3F2

)gd5x

= 12

g(R 12

()2 14e3F2

)gg

d5x. (39)

If we combine S1 and S2, group variational variable and set the variation of the action to zero, we havethe following equations:

0 =

ggRd5x. (40)0 = 1

2

ge3FFd5x (41)0 =

g [14

3F2 1

2()

]d5x (42)

0 =

g [R 12g 1

2

(()() 1

2()2g

) 1

2

(F2

1

4F2g

)]gd5x, (43)

whereF2 = FF . (44)

Equation 40 becomes an integral over the boundary and vanishes [14]. The remaining three integrals defineour field equations [6, 7, 2, 4, 13, 15, 10, 5]:

R 12Rg =

1

2

( 1

2()2g

)+

1

2

(F2

1

4F2g

), (45)

0 = (e3F

), (46)

and 2 = 3

3

2e3F2. (47)

People sometimes eliminate the 12Rg term in equation 45 by subtracting the appropriate multiple of thetrace to get [6]:

R =1

2+

1

2e3

(F2

1

4F2g

). (48)

This is looking pretty familiar! If we let the energy momentum tensor be

T =1

2

( 1

2()2g

)+

1

2

(F2

1

4F2g

), (49)

then equation 45 simply represents the field equations from Einstein gravity [12]. Similarly, although theresa re-scaling by , equation 46 looks very reminiscent of the source-free Maxwell equations in tensor notation[12]. (The other equation from electromagnetism, F = dA, is encoded in our definition of F .) Equation 47is new, though. This is the source term for the scalar field.

Now we can see why = 0 is forbidden in general. Because the scalar and electromagnetic fields interact, = 0 if and only F2 = 0. In other words, if = 0, the entire system just reduces to four-dimensionalEinstein gravity with no electromagnetic field at all. Since we cant get rid of , lets give it a name. Wecall it the dilaton [6].

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2.2 Symmetries

Out of a five-dimensional, purely gravitational system, we seem constructed the field equations for a four-dimensional system that includes gravity, electromagnetism, and a scalar field. If this is truly the case,however, the symmetries of the five-dimensional spacetime have to translate into the symmetries of a four-dimensional spacetime and the gauge-invariances of the vector potential and scalar field [6, 7]. Lets see ifwe can understand how these symmetries appear. As before, this derivation closely follows that of Pope [6].

The five-dimensional Einstein theory is invariant under five-dimensional general coordinate transforma-tions, which can be written infinitesimal form as

xM = M (x), (50)where M is an arbitrary function of all five coordinates [6]. This induces a change in the five-dimensionalmetric [6]:

gMN = PP gMN + gPNM

P + gMPN P . (51)

However, Kaluzas cylinder condition is much more restrictive. It enforces that our coordinate system beClairaut and that 4 be a Killing vector (see section 2.1.1). Thus, in Kaluzas theory, our arbitrary functionA takes on a particular form:

(x) = (x) and z(xA) = cx4 + (x), (52)

where (x) is a general coordinate transformation on M(4), c R is a constant, and (x) is an arbitrarydifferentiable function onM(4) [6]. These are the coordinate transformations allowed in the five-dimensionaltheory.

We can calculate the effects of an allowed coordinate on the four-dimensional metric, vector potential,and dilaton field by plugging the allowed transformations into equation 7. As it turns out, almost all of thefour-dimensional symmetries were looking for can be attained with c = 0. For simplicity, we will discussthis case first. Then well discuss the symmetry embodied by c.

First, lets look at g44:

g44 = PP g44 + gP44

P + g4P4xiP

= g44 + 2cg44

e2 = e2 = . (53)

This implies that does indeed transform as a scalar under four-dimensional general coordinate transfor-mations. We also see the first hints of gauge symmetry: is independent , which will be the gauge choicefor our vector potential. These are the symmetries we would expect for a scalar field [6].

Similarly,

g4 = g4 + g4

e2A = e2A + e2A A = A +A + (x). (54)

The A and A terms are appropriate for covector on M(4), while the term reflects theappropriate symmetry for a U(1) gauge field with gauge parameter . All this tells us that A behavesappropriately for the electromagnetic vector potential [6].

Finally,

g = g + g

+ g + g4

4 + g44

g = g + g + g. (55)This is, of course, the appropriate change in the metric under a general coordinate transformation. Themetric is thus independent of the gauge parameter , as it ought to be if A is Maxwells vector potential[6].

8

With c = 0 weve found all the symmetries of an Einstein-Maxwell system: covariance with a generalcoordinate transformation and gauge invariance under . However, were still missing the symmetry asso-ciated with a scalar field; only the change in the dilaton should matter, not its overall value. This is theconstant shift symmetry [6]. To see how this symmetry appears, we need to discuss one further symmetryof the five-dimensional theory.

In addition to general coordinate transformations, we can rescale the five-dimensional metric by a posi-tive6 constant factor [6],

gMN 2gMN , R. (56)The Riemann tensor is inert under this transformation. However, when we contract its indices to form theRicci scalar, we pick up one over the scale factor from the inverse metric [6]:

R 2R. (57)

This rescales the Lagrangian density by a constant factor. However, we can divide it away when we set thevariation of the action to zero and it has no effect on the field equations [6].

Now lets see what effect this rescaling has on the lower-dimensional theory. We can write this symmetryinfinitesimally as

gMN = 2agMN , (58)

where a is some constant parameter. Lets group this symmetry with the symmetry induced in the metricby a nonzero c:

gMN = c4M g4N + c

4N gM4 + 2agMN , (59)

where AB is the Kronecker delta [6]. If we plug the five-dimensional line element (7) into this, we find that

= a+ c, A = cA, and g = 2ag 2g. (60)

This means that, if we rescale the five-dimensional metric, we can allow c to behave as the dilaton shift andkeep the four-dimensional metric inert [6]. To do this, we enforce that our five-dimensional metric rescalesas

a = c3. (61)

If we rescale when we choose a new set of general coordinates, then the c transformation becomes [6]:

= 2

3

3c, A = cA, and g = 0. (62)

This is exactly what wed expect for a dilaton shift of 23c/3. Thus, with the additional constraint onmetric rescaling, the system is symmetric under this transformation too [6].

So, if we impose Kaluzas cylinder condition on five-dimensional Einstein gravity, we recover field equa-tions consistent with a four-dimensional system that contains Einstein gravity, Maxwell electromagnetism,and a scalar field. Furthermore, transformations in the higher-dimensional space that preserve the cylin-der condition and the five-dimensional field equations result in the appropriate symmetries in the lower-dimensional spacetime.

3 Compactification and Other Tricks

Even with all the success of Kaluzas theory, the cylinder condition is a bit hard to swallow. With nojustification, it seems rather contrived. Indeed, if we impose an isometry on the higher-dimensional theory,it is perhaps not surprising that this manifests as a gauge symmetry in the lower dimensional theory. Both

6In theory, the scale factor could be negative. However, this would reverse our sign convention for spacelike and timelikevectors.

9

the higher-dimensional isometry and the lower-dimensional gauge symmetry are manifestations of a one-dimensional Lie symmetry group, after all. It would therefore be nice if we could somehow justify Kaluzasmysterious condition.

The first successful attempt to justify the cylinder condition comes from Klein [2, 7, 13, 15], and this iswhere the name Kaluza-Klein theory originates. Klein discovered that, if the fifth dimension is a circle, thecircumference can be made so small that any dependence of the metric on x4 is unobservable [2, 7, 13, 15].In modern notation, we say Kleins ansatz is that

M(5) =M(4) U(1), (63)

where U(1) is the circle group [6, 7]. Thus the fifth dimension is compact and we call Kleins schemecompactification [7]

With this topology for the five-dimensional spacetime, the metric must be periodic in x4. This meansthat we can Fourier expand the five-dimensional metric as

gMN (x, x5) =

n=

g(n)MN (x)e

inx4/L, (64)

where x are the coordinates ofM(4), n is the Fourier mode number, and L is the length scale of U(1) [6]. Asusual, higher-|n| modes correspond to higher energy scales. In fact, the n = 0 modes correspond to masslessfields, while the n 6= 0 modes correspond to massive fields [6]. This is easiest to see by studying a simplermodel, elegantly explained by Pope [6].

Suppose we have a massless scalar field on M(5), call it .7 Thanks to the Klein-Gordon equation, wehave that

2 = 0, (65)

where 2 = MM [6]. If we Fourier expand , we find that

(x, x5) =

n=

n(x)einx4/L, (66)

as with the metric [6]. If we apply the five-dimensional Lapplacian operator, we find that the each four-dimensional field satisfies

2n =n2

L2n, (67)

which is the Klein-Gordon equation for a scalar field with mass |n|/L [6]. Similar reasoning holds for theFourier expansion of the metric tensor.

Klein saw the parallels between the periodicity of the compactified dimension and the azimuthal directionin an electrons orbit around a nucleus [2, 13, 15]. He rather ambitiously hoped to use the n 6= 0 modes toexplain the quantization of electric charge in the same way that periodic boundary conditions explain theenergy levels in an atom [2, 13, 15]. In Kleins scheme, the nth harmonic of the metric holds an electriccharge of

en = n2piG

piL,

where G is Newtons constant [2, 13, 15]. This would mean that the charge of an electron is

e =2piG

piL.

We can choose L to match the observed electron charge, and it turns out to be about one hundred Plancklengths [2]. In this scheme, the cylinder condition holds only approximately, and it would break down at the

7We use to avoid confusing this field with the dilaton .

10

appropriate length scale [2, 7].8 String theory and M-theory use a similar compactification approach, butwith more compactified dimensions [7, 6].

If we would prefer the cylinder condition to be absolute, we can set L to be even smaller, on the orderof one Planck length, so that the massive modes have such high energies that they can never be observed.In this case the n = 0 mode is the only observable part of the metric, and it satisfies cylinder condition [6].Kleins conserved charge ensures we can do this [13, 15, 2, 6]. Zero charge can never spontaneously becomenonzero charge [6].

An easy way to understand this is that the Fourier basis functions einx4/L are representations of U(1)

[6]. The n 6= 0 mode functions are paired off by complex conjugates. The function corresponding to modenumber n is the complex conjugate of the function corresponding to mode number n. The n = 0 modefunction is thus a representation of the trivial subgroup of U(1). Since its a subgroup, its closed and noamount of multiplying n = 0 modes together will cause us to jump to an n 6= 0 mode [6].

Although Kleins compactification scheme is by far the most popular way to justify the cylinder condition,it is not the only option. (3 + 1)-dimensional projective space can be thought of as a hypersurface in (4 + 1)-dimensional Minkowski space. In 1930, Veblen and Hoffmann used this idea to justify Kaluzas fifth dimensionand the cylinder condition, where the fifth dimension actually reflects the projective nature of spacetime [16].More than thirty years later, Joseph discussed this possibility as well [17].

People have also argued that the cylinder condition is only an approximation, even on accessible en-ergy scales. The biggest advocate of this approach is Wesson, who has explored some of the cosmologicalimplications of such a scheme [18, 19, 8, 9].

4 Concluding Remarks

From a five-dimensional theory of pure gravity, weve attained a four-dimensional system containing gravity,electromagnetism, and a scalar field. Although Kaluzas cylinder condition appears contrived at first, Kleinand others have demonstrated that it can be justified. This is a remarkable accomplishment, and one thathas garnered a great deal of interest over the years.

Of course, Kaluza-Klein theory has some clear weaknesses. Although the theory predicts the masslessscalar dilaton, we havent observed any such particle. Since its coupled to the electromagnetic field, wewould have observed it if it existed. Additionally, the theory gets much more complicated when we addthe strong and electroweak forces. Each force adds at least one dimension subject to its own cylindercondition, and the whole theory becomes highly nontrivial.

Despite these difficulties, Kaluza-Klein theory remains seductive. Kaluzas geometric approach to unifica-tion appeals to our aesthetic sensibilities and Kleins contribution, the compactification, offers the tantalizingpossibility of explaining quantum mechanics geometricallythe whole thing hints at a theory of quantumgravity. This is certainly why string theory and M-theory use compactified dimensions. It is highly likelythat this beautiful theory will continue to inspire new ideas in and new approaches to gravitational physics.

Acknowledgments

Id like to thank Michael Schmidt for the proof-reading he performed. Id also like to thank Professor OliverDeWolfe for suggesting I start my foray into Kaluza-Klein Theory with Overduin and Wesson [7] and Pope[6].

8Although this energy scale is very much inaccessible at the moment... and probably forever.

11

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[17] J.T. Joseph. Coordinate covariance and the particle spectrum. Physical Review, 126(1):319323, 1962.

[18] P.S. Wesson. A new approach to scale-invariant gravity. Astronomy and Astrophysics, 119:145152,1983.

[19] P.S. Wesson. An embedding for general relativity with variable rest mass. General Relativity andGravitation, 16(2):193203, 1984.

12

Introduction and Some HistoryKaluza's Five-Dimensional SpacetimeGeneral ConstructionThe Cylinder ConditionThe MetricLagrangian DensityField Equations

Symmetries

Compactification and Other TricksConcluding Remarks